If English cricket is about to experience a bright new dawn, the first few rays may have pierced the clouds. Alastair Cook's first three-figure score since a tour game in November at the start of an ill-starred Ashes campaign was a moment to savour, not least because it came on his home ground, as the sun shone. His celebration, cradling his arms in a baby-rocking gesture towards the balcony to mark having recently become a father, added a somewhat clumsy emphasis to the sense of rebirth.

Cook has had plenty of time to sit and analyse the winter passed, his second as England captain. But the strength of his leadership has always been what he does with the bat in hand and the days between his final ODI appearance against Australia on January 26 and the start of Essex's season must have ticked by slowly.

The arrival of his daughter, born at the start of the month, would have helped distract him, not to mention the lambing season. Now he can get back to run scoring. Having harvested a half-century against Cambridge in the tranquil setting of Fenner's last week, he sharpened his focus for the sort of remorseless accumulation he is usually associated with. This was his 46th first-class hundred, 15 innings after his last, and the closest he came to giving a chance was when a top-edged cut flew past slip after he had passed the landmark.

He ended the day unbeaten on 139, perhaps a few short of official "daddy" status but only in strict, Goochian terms. Cries of "Well done, Cookie" were followed by a warm ripple of applause as he made his way off the ground.

Never mind country, this was just the innings his county needed. Essex boast a squad studded with England internationals past, present and possibly future, which makes them a tempting pick when the pre-season promotion prognostications are made, but they did not give a good account of themselves with the bat on the opening day.

Cook happy to be back on the job

Alastair Cook said he was pleased to have been able to get straight back to work after the birth of his first child, Elsie. Less than two weeks after becoming a father, Cook made his first century in five months, with paternity leave shunned in favour off leaving outside off.

"My wife has been very good to me, she's been excellent," he said. "Fingers crossed baby Elsie is sleeping okay and feeding okay. There's not so much a man can do early on. I just wanted to bat and the baby was 10-12 days late and you can't really control that."

While Cook's wife Alice was left in charge of the baby, the England captain took charge of Essex's Championship match against Derbyshire. He described his baby-rocking gesture as a bit of mickey-taking because "I've seen everyone else do it".

He said it was "nice to contribute" a big score after a difficult winter but wasn't about assume his job was complete. "I think it's been a few months without scoring a hundred, any batter is judged on scoring hundreds and it's nice to do that. My job's not done yet."

When James Foster won the toss and decided to make first use of the pitch, the plan would have been to make Derbyshire sweat under a brazen April sun. Fast-forward 24 hours and Cook did just that - in fact, it almost felt as if a few months had flashed by, with two spinners toiling away on a seemingly becalmed surface. Paul Grayson, who left the team to their own devices on Sunday in order to run the London marathon, would have felt the spring returning to his step even as the feeling came back to his toes.

Most of Cook's shots drew an appreciative murmur from the members' stands but one off drive during the afternoon session stood out. Tim Groenewald, who had recently broken a 103-run partnership for the second wicket, overpitched a fraction and Cook snapped like a bear trap, crisply driving four through mid-off. He rattled into the 90s with shots either side of the wicket off a tiring Mark Footitt and reached three figures with a square push against Groenewald. The last time he experienced the sensation was against a lightly equipped Australia A attack in Hobart.

There were moments, particularly early on, when he had to grind out the runs (when are there not, some might say). Cook's familiar shuffle across his stumps and reassuring tuck to leg were not quite in sync and on more than one occasion he came close to tickling through to the keeper as he fell to off, in the manner of his first-innings dismissal. But initially in partnership with Tom Westley, who played some flowing strokes in his 56, and then Ravi Bopara, who fell just short of his own fifty in another century stand, Cook dragged Essex up by the bootlaces. His next, weightier, task is to do the same with England.

Derbyshire were reduced to the status of pliant co-conspirators in a greater drama. Having dismissed Essex for 94 in the first innings and begun the day 45 runs ahead with five wickets in hand, they appeared to have a strong hold on the game. "Who ever hoped like a cricketer?" wrote RC Robertson-Glasgow. Hope appeared to be all Essex had, though David Masters shrugged his broad shoulders at the suggestion. By the close, the game had turned around to the extent that Derbyshire were the team looking for something to cling to.

"There's always money in the banana stand," goes one of the jokes from the US TV series Arrested Development. The Bluth family, in a perpetual state of crisis, set symbolic store by the frozen banana business that abides even as grander schemes collapse around them. For Essex, who have experienced a degree of self-inflicted hardship in recent years, Masters remains as reliable as the banana stand; unflashy but never unloved and always a banker.

It was Masters, a week before his 36th birthday, who provided the early inspiration for Essex's fight back. "If we can bowl like we have today I'm sure they won't get too many more," he said on Sunday. They did not get too many more as Masters followed word with deed, taking three wickets in his opening two overs as Derbyshire lost 5 for 15 on the second morning. His reward was the 21st five-wicket haul of an Essex career that only began in 2008.

Shivnarine Chanderpaul stood aloof from the commotion, attempting to retain the strike in a familiar, crabby rear-guard. When the innings finished, it was hard to say Chanderpaul was, unbeaten on 75 and with no one to accompany him on the dance floor any longer. But it was to be Cook who had the last waltz.

If English cricket is about to experience a bright new dawn, the first few rays may have pierced the clouds. Alastair Cook's first three-figure score since a tour game in November was a moment to savour,